Minimum attaining operators on reducing subspaces: Spectral structure and density
Abstract
In this article, we introduce and study the class ℳr(H) of minimum attaining operators on a separable Hilbert space H whose restrictions to every nonzero reducing subspace are minimum attaining. This class lies between the classes of absolutely minimum attaining operators and minimum attaining operators. We establish a spectral characterization of positive operators in ℳr(H): a positive operator T belongs to ℳr(H) if and only if T=∑λ∈ΛλPλ, where Λ⊆[0,|T|] is a countable well-ordered set, (Pλ)λ∈Λ is a family of mutually orthogonal nonzero projections, and ∑λ∈ΛPλ=I in the strong operator topology. We also establish a connection between ℳr(H) and the class of operators attaining their norm on every nonzero reducing subspace. Further, we obtain stability results under suitable finite-rank and compact perturbations, and prove that ℳr(H) is norm dense in ℬ(H). We also establish the norm density of operators in this class having a nontrivial invariant half-space. Finally, we characterize normal operators in ℳr(H), obtain a corresponding characterization for multiplication operators on L2(𝕋), and characterize the condition ℛT=ℛ|T| in terms of the von Neumann algebra generated by |T|, showing that this condition forces T to be normal.
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BibTeX
@misc{puspendu2026minimum,
title = {{Minimum attaining operators on reducing subspaces: Spectral structure and density}},
author = {Nag Puspendu and Golla Ramesh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.19286},
eprint = {2608.19286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.19286},
}
RIS
TY - UNPB
TI - Minimum attaining operators on reducing subspaces: Spectral structure and density
AU - Nag Puspendu
AU - Golla Ramesh
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2608.19286
UR - https://arxiv.org/abs/2608.19286
AB - In this article, we introduce and study the class ℳ_r(H) of minimum attaining operators on a separable Hilbert space H whose restrictions to every nonzero reducing subspace are minimum attaining. This class lies between the classes of absolutely minimum attaining operators and minimum attaining operators. We establish a spectral characterization of positive operators in ℳ_r(H): a positive operator T belongs to ℳ_r(H) if and only if T=∑_λ∈ΛλP_λ, where Λ⊆[0,|T|] is a countable well-ordered set, (P_λ)_λ∈Λ is a family of mutually orthogonal nonzero projections, and ∑_λ∈ΛP_λ=I in the strong operator topology. We also establish a connection between ℳ_r(H) and the class of operators attaining their norm on every nonzero reducing subspace. Further, we obtain stability results under suitable finite-rank and compact perturbations, and prove that ℳ_r(H) is norm dense in ℬ(H). We also establish the norm density of operators in this class having a nontrivial invariant half-space. Finally, we characterize normal operators in ℳ_r(H), obtain a corresponding characterization for multiplication operators on L^2(𝕋), and characterize the condition ℛ_T=ℛ_|T| in terms of the von Neumann algebra generated by |T|, showing that this condition forces T to be normal.
ER -